The breakdown point behavior of $M$-estimators in linear models
with fixed designs, arising from planned experiments or qualitative factors, is
characterized. Particularly, this behavior at fixed designs is quite different
from that at designs which can be corrupted by outliers, the situation
prevailing in the literature. For fixed designs, the breakdown points of robust
$M$-estimators (those with bounded derivative of the score function), depend on
the design and the variation exponent (index) of the score function. This
general result implies that the highest breakdown point within all regression
equivariant estimators can be attained also by certain $M$-estimators: those
with slowly varying score function, like the Cauchy or slash maximum likelihood
estimator. The $M$-estimators with variation exponent greater than 0, like the
$L_1$ or Huber estimator, exhibit a consider-ably worse breakdown point
behavior.